Evaluate ∫ 5 2 t 2 ln ⁡ ( 3 t 2 ) d t.

Integrate by parts using the formula&nbsp;∫udv=uv-∫vdu, where&nbsp;u=ln(3t2)&nbsp;and&nbsp;dv=52t2.ln(3t2)(56t3)-∫56t3(-2t)dtSimplify.ln(3t2)(5t3)6-∫-5t23dtSince&nbsp;-1&nbsp;is constant with respect to&nbsp;t, move&nbsp;-1&nbsp;out of the integral.ln(3t2)(5t3)6--∫5t23dtSimplify.ln(3t2)(5t3)6+∫5t23dtSince&nbsp;53&nbsp;is constant with respect to&nbsp;t, move&nbsp;53&nbsp;out of the integral.ln(3t2)(5t3)6+53∫t2dtBy the Power Rule, the integral of&nbsp;t2&nbsp;with respect to&nbsp;t&nbsp;is&nbsp;13t3.ln(3t2)(5t3)6+53(13t3+C)Simplify the answer.56ln(3t2)t3+59t3+C

Expert Answer to "Evaluate ∫ 5 2 t 2 ln..."

Carina Valenzuela

Answered question

2022-04-10

Evaluate

\int \frac{5}{2} ﻿ t^{2} \ln (\frac{3}{t^{2}}) d t

Answer & Explanation

Calvin Oneill

Beginner2022-04-11Added 20 answers

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (\frac{3}{t^{2}})$ and $d v = \frac{5}{2} t^{2}$ .

$\ln (\frac{3}{t^{2}}) (\frac{5}{6} t^{3}) - \int \frac{5}{6} t^{3} (- \frac{2}{t}) d t$

Simplify.

$\frac{\ln (\frac{3}{t^{2}}) (5 t^{3})}{6} - \int - \frac{5 t^{2}}{3} d t$

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$\frac{\ln (\frac{3}{t^{2}}) (5 t^{3})}{6} - - \int \frac{5 t^{2}}{3} d t$

Simplify.

$\frac{\ln (\frac{3}{t^{2}}) (5 t^{3})}{6} + \int \frac{5 t^{2}}{3} d t$

Since $\frac{5}{3}$ is constant with respect to $t$ , move $\frac{5}{3}$ out of the integral.

$\frac{\ln (\frac{3}{t^{2}}) (5 t^{3})}{6} + \frac{5}{3} \int t^{2} d t$

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$\frac{\ln (\frac{3}{t^{2}}) (5 t^{3})}{6} + \frac{5}{3} (\frac{1}{3} t^{3} + C)$

Simplify the answer.

$\frac{5}{6} \ln (\frac{3}{t^{2}}) t^{3} + \frac{5}{9} t^{3} + C$

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